U Zʹjr@sddlmZddlmZddlmZddlmZmZm Z ddl m Z ddl m Z ddlmZmZddlmZmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddl m!Z!ddl"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2edZ3edZ4ddZ5ddZ6ddZ7ddZ8ddZ9ddZ:dd Z;d!d"Zd'd(Z?d)d*Z@d+d,ZAd-d.ZBd/d0ZCd1d2ZDd3d4ZEd5d6ZFd7d8ZGd9d:ZHe!d;d<ZId=S)>)randint)Function)Mul)IRationaloo)Eq)S)Dummysymbols)explog)tanh)sqrt)sin)Poly)ratsimp) checkodesol)slow)riccati_normalriccati_inverse_normalriccati_reduced match_riccatiinverse_transform_poly limit_at_infcheck_necessary_conds val_at_infconstruct_c_case_1construct_c_case_2construct_c_case_3construct_d_case_4construct_d_case_5construct_d_case_6rational_laurent_series solve_riccatifxcCstt| |td|S)N)rrmaxintr*c/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/solvers/ode/tests/test_riccati.py rand_rationalsr,cs tfddt|dD|S)Ncsg|] }tqSr*)r,).0_r(r*r+ "szrand_poly..r')rrange)r&degreer)r*r(r+ rand_poly!sr2cCsPtd|}td|}t|||}t|||}|td|krHt|||}q,||S)Nr'r)rr2r)r&r1r)ZdegnumZdegdennumdenr*r*r+rand_rational_function%s    r5c Cs|}||}t|dd}t|dd}|dkrZ normal_eqr*r*r+test_riccati_reducedsf 0<8>   :@  (FB4@( rWc Cstttdtddtddtddtdd tdd tdd td dttddtd td dttttd ddtdddtddtddtddtddtddtddtddtddtddtddtdtddtddtddtddddtdd tdd tdd td tdddtddd tdd dtd ftttdtdtdd ttddtdd ttdtdtd!dd"tdd#td$d%tdd&tddd'tdtd!dddtddtdd(d)tddtd"d%td&tdd*tdddd tdtdd ftttd+td,d-tdd.tdd/tdd0td1d)tdd&td,d2tdd3tdd4tdd1ttd5d ttddtttdtddtddttddtdd+tdd)td,d&tdd2tdd3tdd4td1d-tdd)td,d&tdd2tdd3tdd4td1d.tdd)td,d&tdd2tdd3tdd4td1d6td7td,d tdd8tdd9tdd:td)td5d d7dtddtd,d;tdddtd,dtdd;tddresZb0rRrSmatchfuncsr*r*r+test_match_riccatisl8  .D.    :  >  :.   @8.(       @<8<@@ Przc CsXtdtddtddtdttdtdtdd tdd td dtddtd d td d tddtdd tdtd ftd ttdtd dtddtddtdtd ftdtddtddtdttdtddtddtdtdftdtddtd d tddtd dtd tdtdd tttdtdttdftd tddtd d td dtdtdd td ttdtd dtddtddtdtdfg}|D]"\}}}t||t|ks0tq0dS)a This function tests the valuation of rational function at oo. Each test case has 3 values - 1. num - Numerator of rational function. 2. den - Denominator of rational function. 3. val_inf - Valuation of rational function at oo rhr6rIr7 rLirKrBrArG r'rMrCrJrirNN)rr&rr9)rPr3r4valr*r*r+test_val_at_infs.(x4((XH4rcCsftddddgdksttddddgdks0ttddddgdksHttdddd d gd ksbtd S) zt This function tests the necessary conditions for a Riccati ODE to have a rational particular solution. rVr'r7rGFr6rLrIrJTN)rr9r*r*r*r+test_necessary_condsFsrc Cs>dtddtddtddtddtdd td d tdd tdd td dtddtddtddtddtd dtdd tdd td d td d tdd tdd td d tddtddtddtd d tddtd dtddtd d tddtddtd dtddtd dtddtddtddtd dtddtddtdg}|D]N}dd|D\}}t||t\}}|tdt||kstqdS)zi This function tests the substitution x -> 1/x in rational functions represented using Poly. r~r6rIr7rLr[rBrurGPrfKrJirlrArgradrhr]rKcSsg|]}t|tqSr*)rr&)r-er*r*r+r/gsz/test_inverse_transform_poly..r'N)r&Zas_numer_denomrsubsrOr9)Zfnsr%r3r4r*r*r+test_inverse_transform_polyUs .^j^t  rc Cstdtddtdttdtddtddtdtdftdtd dtdd tddtd ttd tdd tddtdttftdtdd tddtddtddtd dtddtddtdttdtddtddtddtd dtdd tdd!td"tt ftd!tdd#tdd$tdd$td d%tdd&tdd'td(ttd)tddtdd*tdd*td d+tdd,tdd-td.ttd/dftd0td d1tdd.tdd2td3ttd4td dtdd5tddtdttd0d4fg}|D]"\}}}t||t|kstqd6S)7a This function tests the limit at oo of a rational function. Each test case has 3 values - 1. num - Numerator of rational function. 2. den - Denominator of rational function. 3. limit_at_inf - Limit of rational function at oo r7ra r6r\rrGiixii'ii!rIrAirLrerBiXHi ivii| Tir_irfrr]rrr~iii i i ir' iiZi_iN)rr&rr rr9)rPr3r4Zlimr*r*r+test_limit_at_infls:(4(TXXX 44 rc Cstdtddtddtdtddtdtdd td d tdd tdd tdd tdtddtdtddtd td gtddtd td ggftdtddtddtdtddtdtddtddtddtddtdtddtd dtddtddgtddtddggftdtdtddtdtdddtdtdddtdtddd tdtdtdd tdd tdtddtddttddtddd!d"d#tdd$ddgtddttddtddd!d"d#tdd$ddggfg}|D]&\}}}}t||t||kstqd%S)&ab This function tests the Case 1 in the step to calculate coefficients of c-vectors. Each test case has 4 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. c - The c-vector for the pole. rVr6r7rGrBT extensionrIrArKrJrLrr'iii0i0iiiei1ijizrrr|z QQdomainFrD,N)rr&r rrrrr9)rPr3r4polecr*r*r+test_construct_c_case_1s6,P:,D 2P BBrcCstdtddttddtdtddddt dtdgtdtdggftdtddtd d tddtddtdtddtddtddtdddtd  d gtd  d ggfttdtdd ttddttdd tddtdddd d tdtddd tdd ddtdd gdtdtddd tdd ddtdd ggftdtdtd dtddtd tdd tdtddtd d d dtdtd dtdddtddgdtdtddtdddtd dggfttddtddtdtddtddtddtddddtdtd dtddddtddtdd gd!tdtddtddddtddtd d ggfttddtddtttddtddtddtd"tdd dtd"d#td d td"gtd" tdd dtd"d#td  d td" ggfg}|D]*\}}}}}t||t|||kstqd$S)%a This function tests the Case 2 in the step to calculate coefficients of c-vectors. Each test case has 5 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. mul - The multiplicity of the pole. 5. c - The c-vector for the pole. r'Trr7rFr6rBrJrGrAYrbrKrfiWrNri7i8irLr^B6i$rviriirN)rr&rr rrr9)rPr3r4rmulrr*r*r+test_construct_c_case_2s^ $0$   ::   44 $ BB4:,rcCstdggkstdS)z` This function tests the Case 3 in the step to calculate coefficients of c-vectors. r'N)rr9r*r*r*r+test_construct_c_case_3srcCsttd dtddtddtdtddtdtddtddtdtddddtd tdd ttd d tddgd td t ddttd d tddggfttd dtddtddtddtddtdtddttddtddtdtdtddddttt dtdgdtt tdtdggftdtdtdtddtdtddtdtddtdtddtdtddddtdtddtdtdtdtdtd ttdddtdtddgdtdtddtdtdtd tdtdttdddtdtddggftd tddtddtdtddtddtdtdtddddtddtdtdt td d tgd!tdd"tdt dttd d tggfttd tdtdtdttddttdtdddd#tddtt tt d$dtdgdtddttt td$dtdggfttd tddtddtdtdtddtdtddtdtdtddddtdtddtdtdtdtdtdtdtd ttd d%dtdtddgd tdtdd tdtdtd tdtd tdtdttd d%dtdtddggfg}|D]8\}}}}t||tt|d}t||d|ks^tq^d&S)'a9 This function tests the Case 4 in the step to calculate coefficients of the d-vector. Each test case has 4 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. mul - Multiplicity of oo as a pole. 4. d - The d-vector. rBr7rGr6TrrKrhr^rVrZrrLrArJrMrorNi rirliIBi ir')rg;iirrCr}rIN) rr&rr rr#rr r9)rPr3r4rdserr*r*r+test_construct_d_case_4sl6,02N06D H4. 822 .XNXD0.rc Cstdtdtdtdtddtdtddtddtdtddtddtd dgtd dtddggftdtdtd tdtddtdtd d tdtdd td dtddtddtd td d tddd tddgtd ddtddggfttdtdtd d tdtdd tdtd d tdddtddgtd ddtddggfg}|D]0\}}}t||ttdd}t||kstqdS)a This function tests the Case 5 in the step to calculate coefficients of the d-vector. Each test case has 3 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. d - The d-vector. r7r6TrrKrBr'rnrGZZrrArNr^rCrN)rr&rr#rr!r9)rPr3r4rrr*r*r+test_construct_d_case_5Es $,28D8 8rcCstdtddtddtdtddtddtdtddtddtdgtddtdggftdtd dtddtdtddttd tddtddtd dtddtd tdddgd ggftd td tddtdtddtd tddtddtd dtddtddtd dtddtdtdddgd ggfg}|D]"\}}}t||t|kstqdS)a This function tests the Case 6 in the step to calculate coefficients of the d-vector. Each test case has 3 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. d - The d-vector. rNr7rBrrrGrhr'r6rLrKrrCr|rJrIrcrAr^r."*N)rr&r rr"r9)rPr3r4rr*r*r+test_construct_d_case_6fs, ,*,H (T rcCsttddtdtddttdttddtddddd dd dd d ftd tdd tdtddtd tddtddtddtdtddtddddtddtddtddtdddftdtddttddtddtddtdd tdd!dtdtdddtdtdtddtddddtdd"dtdtd#d"dtdd$d%d#tdd"dtdd#tddtd#d"dtdd$d%d#tddd"dtdd#tddd&fttddtddtdd'td(tddttddtddtdddd)ddd*d+d,ftdtddtddtdtddtdtdtddtddtdtddtddd)dtd-d.td/d0td1d2td3d4td5d6d7ftd8tddtdtddtdtddtddtddtddtdtddtd)dd)d)td9 d:d)d#td;dd<fg}|D].\}}}}}}|t||t|||kstqd=S)>a This function tests the computation of coefficients of Laurent series of a rational function. Each test case has 5 values - 1. num - Numerator of the rational function. 2. den - Denominator of the rational function. 3. x0 - Point about which Laurent series is to be calculated. 4. mul - Multiplicity of x0 if x0 is a pole of the rational function (0 otherwise). 5. n - Number of terms upto which the series is to be calculated. r7r6rKTrr'rLrAir})r'rrFrNrVrH@iirGri?iirIiiKiil`B l-Ri0i}')rr7rFr'rBrHrJrrVrFFrD)rGr6r7r'rrFrhr{rr~ro)r6r7r'rrFrNi i ii)l?PTl@qslF41l@4lJ!tzlZdj m)rrFrNrVrHrGrr|)rrNrCrFrVrHN)rr&r rrrr#r9)rPr3r4Zx0rnrr*r*r+test_rational_laurent_seriess 8   P2 4 ,4   D"/rcst|tr|j|j}t|tt\}}ttttf|}tdfdd|D}t ddt ||Dsrt t fddt ||Dst dS)z Helper function to check if actual solution matches expected solution if actual solution contains dummy symbols. C1csg|]}|qSr*)r)r-r?r dummy_symr*r+r/sz#check_dummy_sol..css|]}|dVqdS)rNr*)r-r&r*r*r+ sz"check_dummy_sol..c3s|]\}}||VqdS)N)Zdummy_eq)r-s1s2)rr*r+rsN) isinstancerlhsrhsrr%r&r$r allrr9zip)r>Zsolserr.ryZsolsr*rr+check_dummy_sols  rcCs0 td}ttttttdddttttdttttd gfttdtttdtttdtdtttd|t|ttdgfdtdttttdtttttdttdttttt|dtd|tgfttttttd dtdtdtttdtdtgftddtdtttttdtttttt|ttddt|tdttttgftdttttdtdttdtttttttt|tdt|tdttttdgfttd tttdtdd td tdddtddtttd |td |dtd dtddtddtddtd d |tdd |td|d td dtd dtddtddtdd ttttdtddtddtdgfttdtttdtd dtd dtddtdd tdd tddtdtttdtd dtdtddtddtddtddtdgftttttd  dtddtddtddtdtddtddtddtd dtddtd d td dtd dtddtddtddtdttdtttttdtd d td dtdd tddtdd tdgftttttttdtdtdttddtddtdd td!d"tddtddtddtttddtdtdgfttttd#tdd$tddtddtddtddttdd tdtttdtddtdgftttdtddttdtd tdtdttttddtddtdttddttt| tdtddt|t|tdtdtdttttd%tdgftttdtttddttttt |ttd|tdtttt gfttttttttdddttdtddttddtdtddtd dd&tdd'td(d"tddtdtttd tttttdtd)d*td+d,tdd-tdd.tdd/td d0tdd1td d2td d3td d4tdd5tdd6tdd7td8dtd)d9td+d:tdd;tddtd d?td d@td dAtddBtddBtddCtgfttttdDtddDtdt dtddttdd tttd tgfttttdEtdddtdtdtddttdd dttttdttttdEtddgfg}|D]\}}t||| qdFS)Ga This function tests the computation of rational particular solutions for a Riccati ODE. Each test case has 2 values - 1. eq - Riccati ODE to be solved. 2. sol - Expected solution to the equation. Some examples have been taken from the paper - "Statistical Investigation of First-Order Algebraic ODEs and their Rational General Solutions" by Georg Grasegger, N. Thieu Vo, Franz Winkler https://www3.risc.jku.at/publications/download/risc_5197/RISCReport15-19.pdf C0r7rrGrNr'r6r~rarArKrLrBrfr`r\r9:rIrJrur]rgr|rrhiircrr0rFirrrrr{rri0HiiFi4"i<ii2ҍi"0 iimi^ri ir?r*r*r+test_solve_riccatis\"$4&N"08:F6@J:    F  H FN     R D2 "H"J&F(:J .J        F       LHe rcCstd}ttttdttttdddtttddtddtddtdd td d td d tddtddtddtttdtddtdgfg}|D]\}}t|||qdS)z This function tests the computation of rational particular solutions for a Riccati ODE. Each test case has 2 values - 1. eq - Riccati ODE to be solved. 2. sol - Expected solution to the equation. rr'r6r7rJrKii/ii-rIrGi*irrrLN)r rr%r&r8rrr*r*r+test_solve_riccati_slowXs HD" rN)JZsympy.core.randomrZsympy.core.functionrZsympy.core.mulrZsympy.core.numbersrrrZsympy.core.relationalrZsympy.core.singletonr Zsympy.core.symbolr r Z&sympy.functions.elementary.exponentialr r Z%sympy.functions.elementary.hyperbolicrZ(sympy.functions.elementary.miscellaneousrZ(sympy.functions.elementary.trigonometricrZsympy.polys.polytoolsrZsympy.simplify.ratsimprZsympy.solvers.ode.subscheckrZsympy.testing.pytestrZsympy.solvers.ode.riccatirrrrrrrrrrrr r!r"r#r$r%r&r,r2r5r@rUrWrzrrrrrrrrrrrrrrr*r*r*r+sN            H B6c,.&>?! C}